Digital-to-Digital Frequency Transformations 1.1 2005/02/21 01:31:31 US/Central 2005/04/24 16:09:25 GMT-5 Douglas L. Jones dl-jones@uiuc.edu Douglas L. Jones dl-jones@uiuc.edu Jeffrey Silverman jsilv@rice.edu Kyle Clarkson kclarks@rice.edu Given a prototype digital filter design, transformations similar to the bilinear transform can also be developed. Requirements on such a mapping z -1 g z -1 : points inside the unit circle stay inside the unit circle (condition to preserve stability) unit circle is mapped to itself (preserves frequency response) This condition implies that ω 1 g ω g ω g ω requires that g ω 1 on the unit circle! Thus we require an all-pass transformation: g z -1 k 1 p z -1 α k 1 α k z -1 where α K 1 , which is required to satisfy this condition. Lowpass-to-Lowpass z 1 -1 z -1 a 1 a z -1 which maps original filter with a cutoff at ωc to a new filter with cutoff ωc ′ , a 1 2 ω c ω c ′ 1 2 ω c ω c ′ Lowpass-to-Highpass z 1 -1 z -1 a 1 a z -1 which maps original filter with a cutoff at ωc to a frequency reversed filter with cutoff ωc ′ , a 1 2 ω c ω c ′ 1 2 ω c ω c ′ (Interesting and occasionally useful!)